= Signed distance from a point to a plane
{title2=$s=\mathbf n\cdot\mathbf p-d$}
Orient the <plane> $\mathbf n\cdot\mathbf r=d$ with a <unit normal> $\mathbf n$. The signed distance of a point $\mathbf p$ from the plane is
$$
s=\mathbf n\cdot\mathbf p-d.
$$
Its absolute value is the ordinary <distance from a point to a plane>. The closest point in the plane is $\mathbf p-s\mathbf n$, so the displacement from the point to its projection is $-s\mathbf n$. Reversing the normal and $d$ reverses $s$ without changing the geometric plane or the unsigned distance. A sphere of radius $q$ centred at $\mathbf p$ has <sphere-plane intersection> radius $\sqrt{q^2-s^2}$ when $|s|<q$.
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